Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
119 tokens/sec
GPT-4o
56 tokens/sec
Gemini 2.5 Pro Pro
43 tokens/sec
o3 Pro
6 tokens/sec
GPT-4.1 Pro
47 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

On the Benefits of Over-parameterization for Out-of-Distribution Generalization (2403.17592v1)

Published 26 Mar 2024 in cs.LG and stat.ML

Abstract: In recent years, machine learning models have achieved success based on the independently and identically distributed assumption. However, this assumption can be easily violated in real-world applications, leading to the Out-of-Distribution (OOD) problem. Understanding how modern over-parameterized DNNs behave under non-trivial natural distributional shifts is essential, as current theoretical understanding is insufficient. Existing theoretical works often provide meaningless results for over-parameterized models in OOD scenarios or even contradict empirical findings. To this end, we are investigating the performance of the over-parameterized model in terms of OOD generalization under the general benign overfitting conditions. Our analysis focuses on a random feature model and examines non-trivial natural distributional shifts, where the benign overfitting estimators demonstrate a constant excess OOD loss, despite achieving zero excess in-distribution (ID) loss. We demonstrate that in this scenario, further increasing the model's parameterization can significantly reduce the OOD loss. Intuitively, the variance term of ID loss remains low due to orthogonality of long-tail features, meaning overfitting noise during training generally doesn't raise testing loss. However, in OOD cases, distributional shift increases the variance term. Thankfully, the inherent shift is unrelated to individual x, maintaining the orthogonality of long-tail features. Expanding the hidden dimension can additionally improve this orthogonality by mapping the features into higher-dimensional spaces, thereby reducing the variance term. We further show that model ensembles also improve OOD loss, akin to increasing model capacity. These insights explain the empirical phenomenon of enhanced OOD generalization through model ensembles, supported by consistent simulations with theoretical results.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (105)
  1. Invariant risk minimization games. In International Conference on Machine Learning, pages 145–155. PMLR.
  2. Towards understanding ensemble, knowledge distillation and self-distillation in deep learning. arXiv preprint arXiv:2012.09816.
  3. Generalization bounds for (wasserstein) robust optimization. Advances in Neural Information Processing Systems, 34:10382–10392.
  4. Invariant risk minimization. arXiv preprint arXiv:1907.02893.
  5. Implicit regularization in deep matrix factorization. Advances in Neural Information Processing Systems, 32.
  6. Ensemble of averages: Improving model selection and boosting performance in domain generalization. Advances in Neural Information Processing Systems, 35:8265–8277.
  7. Bai, Z. D. (2008). Methodologies in spectral analysis of large dimensional random matrices, a review. In Advances in statistics, pages 174–240. World Scientific.
  8. Benign overfitting in linear regression. arXiv preprint arXiv:1906.11300v3.
  9. Reconciling modern machine-learning practice and the classical bias–variance trade-off. Proceedings of the National Academy of Sciences, 116(32):15849–15854.
  10. Two models of double descent for weak features. SIAM Journal on Mathematics of Data Science, 2(4):1167–1180.
  11. To understand deep learning we need to understand kernel learning. In International Conference on Machine Learning, pages 541–549. PMLR.
  12. Analysis of representations for domain adaptation. Advances in neural information processing systems, 19.
  13. Discriminative learning under covariate shift. Journal of Machine Learning Research, 10(9).
  14. Ensembles for feature selection: A review and future trends. Information Fusion, 52:1–12.
  15. Breiman, L. (1996). Bagging predictors. Machine learning, 24:123–140.
  16. Managing diversity in regression ensembles. Journal of machine learning research, 6(9).
  17. Language models are few-shot learners. Advances in neural information processing systems, 33:1877–1901.
  18. Benign overfitting in two-layer convolutional neural networks. Advances in neural information processing systems, 35:25237–25250.
  19. The implicit bias of batch normalization in linear models and two-layer linear convolutional neural networks. In The Thirty Sixth Annual Conference on Learning Theory, pages 5699–5753. PMLR.
  20. Swad: Domain generalization by seeking flat minima. Advances in Neural Information Processing Systems, 34:22405–22418.
  21. Finite-sample analysis of interpolating linear classifiers in the overparameterized regime. The Journal of Machine Learning Research, 22(1):5721–5750.
  22. Benign overfitting in adversarially robust linear classification. In Uncertainty in Artificial Intelligence, pages 313–323. PMLR.
  23. Dietterich, T. G. (2000). Ensemble methods in machine learning. In International workshop on multiple classifier systems, pages 1–15. Springer.
  24. Learning models with uniform performance via distributionally robust optimization. The Annals of Statistics, 49(3):1378–1406.
  25. El Karoui, N. (2010). The spectrum of kernel random matrices. The Annals of Statistics, 38(1):1–50.
  26. Data-driven distributionally robust optimization using the wasserstein metric: Performance guarantees and tractable reformulations. arXiv preprint arXiv:1505.05116.
  27. Freedman, D. A. (1981). Bootstrapping regression models. The Annals of Statistics, 9(6):1218–1228.
  28. A decision-theoretic generalization of on-line learning and an application to boosting. Journal of computer and system sciences, 55(1):119–139.
  29. Friedman, J. H. (2001). Greedy function approximation: a gradient boosting machine. Annals of statistics, pages 1189–1232.
  30. A review on ensembles for the class imbalance problem: bagging-, boosting-, and hybrid-based approaches. IEEE Transactions on Systems, Man, and Cybernetics, Part C (Applications and Reviews), 42(4):463–484.
  31. Domain-adversarial training of neural networks. Journal of machine learning research, 17(59):1–35.
  32. Domain adaptation with conditional transferable components. In International conference on machine learning, pages 2839–2848. PMLR.
  33. Covariate shift by kernel mean matching.
  34. In search of lost domain generalization. arXiv preprint arXiv:2007.01434.
  35. Characterizing implicit bias in terms of optimization geometry. In International Conference on Machine Learning, pages 1832–1841. PMLR.
  36. Implicit regularization in matrix factorization. Advances in neural information processing systems, 30.
  37. Neural network ensembles. IEEE transactions on pattern analysis and machine intelligence, 12(10):993–1001.
  38. The surprising harmfulness of benign overfitting for adversarial robustness. arXiv preprint arXiv:2401.12236.
  39. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 770–778.
  40. Conditional variance penalties and domain shift robustness. arXiv preprint arXiv:1710.11469.
  41. Kullback-leibler divergence constrained distributionally robust optimization. Available at Optimization Online, 1(2):9.
  42. Causal discovery from heterogeneous/nonstationary data. The Journal of Machine Learning Research, 21(1):3482–3534.
  43. Neural tangent kernel: Convergence and generalization in neural networks. Advances in neural information processing systems, 31.
  44. Gradient descent aligns the layers of deep linear networks. arXiv preprint arXiv:1810.02032.
  45. The implicit bias of gradient descent on nonseparable data. In Conference on Learning Theory, pages 1772–1798. PMLR.
  46. On large-batch training for deep learning: Generalization gap and sharp minima. arXiv preprint arXiv:1609.04836.
  47. Concentration inequalities and moment bounds for sample covariance operators. Bernoulli, pages 110–133.
  48. Dynamic weighted majority: An ensemble method for drifting concepts. The Journal of Machine Learning Research, 8:2755–2790.
  49. Neural network ensembles, cross validation, and active learning. Advances in neural information processing systems, 7.
  50. Wasserstein distributionally robust optimization: Theory and applications in machine learning. In Operations research & management science in the age of analytics, pages 130–166. Informs.
  51. Calibrated ensembles can mitigate accuracy tradeoffs under distribution shift. In Uncertainty in Artificial Intelligence, pages 1041–1051. PMLR.
  52. Kuncheva, L. I. (2014). Combining pattern classifiers: methods and algorithms. John Wiley & Sons.
  53. Large-scale methods for distributionally robust optimization. Advances in Neural Information Processing Systems, 33:8847–8860.
  54. Just interpolate: Kernel “ridgeless” regression can generalize.
  55. Bayesian invariant risk minimization. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 16021–16030.
  56. Spurious feature diversification improves out-of-distribution generalization. arXiv preprint arXiv:2309.17230.
  57. Zin: When and how to learn invariance without environment partition? Advances in Neural Information Processing Systems, 35:24529–24542.
  58. Decomposition algorithm for distributionally robust optimization using wasserstein metric. arXiv preprint arXiv:1704.03920.
  59. Distributionally robust optimization with bias & variance reduced gradients. In The Twelfth International Conference on Learning Representations.
  60. Explicit tradeoffs between adversarial and natural distributional robustness. Advances in Neural Information Processing Systems, 35:38761–38774.
  61. Data-driven distributionally robust optimization using the wasserstein metric: performance guarantees and tractable reformulations. Mathematical Programming, 171(1-2):115–166.
  62. Classification vs regression in overparameterized regimes: Does the loss function matter? The Journal of Machine Learning Research, 22(1):10104–10172.
  63. Harmless interpolation of noisy data in regression. IEEE Journal on Selected Areas in Information Theory, 1(1):67–83.
  64. Stochastic gradient methods for distributionally robust optimization with f-divergences. Advances in neural information processing systems, 29.
  65. Exploring generalization in deep learning. Advances in neural information processing systems, 30.
  66. Path-sgd: Path-normalized optimization in deep neural networks. Advances in neural information processing systems, 28.
  67. In search of the real inductive bias: On the role of implicit regularization in deep learning. arXiv preprint arXiv:1412.6614.
  68. Popular ensemble methods: An empirical study. Journal of artificial intelligence research, 11:169–198.
  69. Pearl, J. (2009). Causality. Cambridge university press.
  70. Moment matching for multi-source domain adaptation. In Proceedings of the IEEE/CVF international conference on computer vision, pages 1406–1415.
  71. When networks disagree: Ensemble methods for hybrid neural networks. In How We Learn; How We Remember: Toward An Understanding Of Brain And Neural Systems: Selected Papers of Leon N Cooper, pages 342–358. World Scientific.
  72. Causal inference by using invariant prediction: identification and confidence intervals. Journal of the Royal Statistical Society Series B: Statistical Methodology, 78(5):947–1012.
  73. Polikar, R. (2006). Ensemble based systems in decision making. IEEE Circuits and systems magazine, 6(3):21–45.
  74. An online method for a class of distributionally robust optimization with non-convex objectives. Advances in Neural Information Processing Systems, 34:10067–10080.
  75. Learning transferable visual models from natural language supervision. In International conference on machine learning, pages 8748–8763. PMLR.
  76. Diverse weight averaging for out-of-distribution generalization. arXiv preprint arXiv:2205.09739.
  77. Rotation forest: A new classifier ensemble method. IEEE transactions on pattern analysis and machine intelligence, 28(10):1619–1630.
  78. Rokach, L. (2010). Ensemble-based classifiers. Artificial intelligence review, 33:1–39.
  79. An investigation of why overparameterization exacerbates spurious correlations. In International Conference on Machine Learning, pages 8346–8356. PMLR.
  80. Shamir, O. (2022). The implicit bias of benign overfitting. In Conference on Learning Theory, pages 448–478. PMLR.
  81. More is better in modern machine learning: when infinite overparameterization is optimal and overfitting is obligatory. arXiv preprint arXiv:2311.14646.
  82. Certifying some distributional robustness with principled adversarial training. arXiv preprint arXiv:1710.10571.
  83. The implicit bias of gradient descent on separable data. The Journal of Machine Learning Research, 19(1):2822–2878.
  84. Distributionally robust optimization and generalization in kernel methods. Advances in Neural Information Processing Systems, 32.
  85. Covariate shift adaptation by importance weighted cross validation. Journal of Machine Learning Research, 8(5).
  86. Telgarsky, M. (2013). Margins, shrinkage, and boosting. In International Conference on Machine Learning, pages 307–315. PMLR.
  87. Trainable projected gradient method for robust fine-tuning. arXiv preprint arXiv:2303.10720.
  88. Benign overfitting in ridge regression. Journal of Machine Learning Research, 24(123):1–76.
  89. Vershynin, R. (2018). High-dimensional probability: An introduction with applications in data science, volume 47. Cambridge university press.
  90. Malign overfitting: Interpolation can provably preclude invariance. arXiv preprint arXiv:2211.15724.
  91. Benign overfitting in multiclass classification: All roads lead to interpolation. Advances in Neural Information Processing Systems, 34:24164–24179.
  92. Binary classification of gaussian mixtures: Abundance of support vectors, benign overfitting, and regularization. SIAM Journal on Mathematics of Data Science, 4(1):260–284.
  93. The marginal value of adaptive gradient methods in machine learning. Advances in neural information processing systems, 30.
  94. Model soups: averaging weights of multiple fine-tuned models improves accuracy without increasing inference time. In International Conference on Machine Learning, pages 23965–23998. PMLR.
  95. Robust fine-tuning of zero-shot models. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 7959–7971.
  96. The power and limitation of pretraining-finetuning for linear regression under covariate shift. Advances in Neural Information Processing Systems, 35:33041–33053.
  97. Explaining the success of adaboost and random forests as interpolating classifiers. The Journal of Machine Learning Research, 18(1):1558–1590.
  98. A survey of autonomous driving: Common practices and emerging technologies. IEEE access, 8:58443–58469.
  99. Understanding deep learning (still) requires rethinking generalization. Communications of the ACM, 64(3):107–115.
  100. Stochastic approximation approaches to group distributionally robust optimization. Advances in Neural Information Processing Systems, 36.
  101. Boosting with early stopping: Convergence and consistency.
  102. Sparse invariant risk minimization. In International Conference on Machine Learning, pages 27222–27244. PMLR.
  103. Ensembling neural networks: many could be better than all. Artificial intelligence, 137(1-2):239–263.
  104. The benefits of implicit regularization from sgd in least squares problems. Advances in neural information processing systems, 34:5456–5468.
  105. Benign overfitting of constant-stepsize sgd for linear regression. In Conference on Learning Theory, pages 4633–4635. PMLR.
User Edit Pencil Streamline Icon: https://streamlinehq.com
Authors (4)
  1. Yifan Hao (28 papers)
  2. Yong Lin (77 papers)
  3. Difan Zou (71 papers)
  4. Tong Zhang (569 papers)
Citations (4)
View on Semantic Scholar

Summary

Essay on "On the Benefits of Over-Parameterization for Out-of-Distribution Generalization"

The paper "On the Benefits of Over-Parameterization for Out-of-Distribution Generalization" addresses the intriguing question of how over-parameterized deep neural networks (DNNs) can maintain robust generalization performance, particularly in out-of-distribution (OOD) contexts. This work is situated in the broader landscape of machine learning, where models are often developed under the assumption of independently and identically distributed (IID) data. However, real-world applications frequently violate this assumption, posing significant challenges to the conventional understanding of model generalization.

Theoretical Insights and Methodological Contributions

This research specifically investigates the role of over-parameterization and its connection to the benign overfitting phenomenon in OOD generalization. The authors paper a ReLU-based random feature model under non-trivial natural distributional shifts, a setup where current theoretical frameworks often fall short. They note that typical results for over-parameterized models either do not apply or contradict empirical evidence in OOD scenarios. The analysis focuses on how benign overfitting estimators maintain zero excess in-distribution (ID) loss while enduring constant excess OOD loss, revealing situations where increasing model parameterization can significantly reduce the OOD loss.

A critical contribution of this paper is the development of an analytical framework that quantifies the excess risk associated with such models. Under the benign overfitting assumptions originally presented by Bartlett et al. (2020), they show that while ID testing loss variance remains minimal due to orthogonal characteristics of long-tail features, variance grows in OOD contexts owing to distributional shifts. The authors present quantitative bounds for ID and OOD excess risks, highlighting that an increase in hidden dimensions can reduce OOD loss by enhancing the orthogonality of long-tail features through higher-dimensional mapping.

Empirical and Analytical Findings

Numerically, the paper reports robust performance increase in OOD settings with model ensembles, attributing successful OOD generalization to potential feature diversification. The authors posit that model ensembles enhance performance by approximating the improvement seen with increased capacity, thereby pushing the boundaries of OOD performance, as previously observed in various empirical studies.

The simulation results are consistent with their theoretical analysis, vividly showing that even as the model attains near-optimal ID performance, the OOD performance can still be significant and is improved with further parameterization and model ensembling. These findings offer a rigorous explanation for empirical observations of larger DNNs performing well under non-trivial distributional shifts, directly contrasting existing theories that suggest increased parameterization could potentially lead to instability under shifts.

Implications and Future Directions

This paper revisits the discussion on over-parameterization's effects, notably suggesting that for OOD scenarios, parameterization may indeed have beneficial implications. It calls into question prior belief systems around over-parameterization, suggesting a nuanced perspective where over-parameterization can serve as an asset rather than a liability. Furthermore, the research implications extend into practical avenues where models must be robust to unforeseen data shifts, such as autonomous systems and global data deployments.

The findings indicate potential future research directions, particularly in better understanding how benign overfitting and feature diversification in ensemble learning translate into robust OOD generalization performance. Additionally, the exploration of causal relationships in distributional shifts, contrasted against adversarial examples, sheds light on more refined and practical robustness assurance techniques.

In conclusion, "On the Benefits of Over-Parameterization for Out-of-Distribution Generalization" provides a pivotal theoretical and empirical investigation into contemporary DNN over-parameterization. It contributes to a burgeoning understanding of how robust generalization under OOD conditions is achievable, offering pathways to more resilient and broadly applicable machine learning models.

File Document Download Save Streamline Icon: https://streamlinehq.com PDF File Document Download Save Streamline Icon: https://streamlinehq.com Markdown
X Twitter Logo Streamline Icon: https://streamlinehq.com

Tweets

Youtube Logo Streamline Icon: https://streamlinehq.com

YouTube